When the subject of 'continuity' and 'canonity' come up, I think some clarification is required. Somehow we've shifted gears and are now debating Eastern VS Western storytelling styles. They're not the same thing.
I believe what was originally being discussed was continuity as in ideas, concepts and plot points within a show (in this case, comedy cartoons) remaining consistent and things and events from previous episodes being brought up again or stuck to in later ones. That's not the same thing as expecting all comedy cartoons to tell a singular finite story with dark elements, an ongoing saga with lots of plot twists and not every episode having a happy ending. Those are 2 different things entirely, and I think it's silly to try and hold Johnny Test up to same standards as Avatar, the Last Airbender.
Now in some cases I agree that there should be a certain degree of consistency within shows. It bugs me a bit when a show will violate its' own ground rules just for the sake of story: for example, one episode of Fairly OddParents depicted Cosmo as having a brother when in previous episodes it had been established that he was an only child, or how in the Aqua Teen Hunger Force episode "Laser Lenses", it was shown that Frylock's laser eyes came from special contact lenses that he wore, when in all of the episodes previous to that it had been established that the jewel on Frylock's back was the source of all of his powers. Those kinds of violations do annoy me a bit; it's like how am I supposed to take the show's universe seriously when the people behind the show don't?
However, other shows have followed continuity and called back earlier episodes and remained consistent. Many examples of this have already been pointed out by others, so the argument that comedy cartoons don't follow continuity isn't entirely true.
I also don't think it's as big as issue as some people here are making it out to be. What we want doesn't necessarily matter; it's what the creators want that's important. If they want to set the rest button at the beginning of each new episode, then that's their prerogative. If we don't like the way they do things, then we always have the option on switching the show off.
Going back to the 'story arcs and sagas' thing for a second, I tend to think that the demand placed on continuity in some cases stems from young adults who feel that shows in which the episodes don't wrap everything up at the end and shows which tell a finite story as opposed to a premise which could conceivably go on forever are somehow more "mature", and therefore by association, the people who watch these shows are also more "mature" for enjoying them. It's the same mentality that leads hardcore gamers to judge every video game which incorporates bright colors rather than just blacks, browns and grays and the objective isn't to shoot everything in sight as "kiddie crap" or hardcore comic book fans who feel every superhero should be brooding and angry and have a troubled past. When in fact this method of storytelling is no more or less mature than shows like SpongeBob Squarepants or Chowder or Back to the Barnyard, it's just different. I'm an adult and I have no problem enjoying shows like Chowder or SpongeBob. 'Simple' and 'light' aren't synonymous with 'just for kids'.
When people say "Well, there's continuity on anime", that's not an entirely accurate statement: for one, the anime which gets imported over to the West only represents about 1% of Japanese animation in total. For every anime which does follow continuity and tell a singular finite story, there are hundreds upon hundreds which don't, so the likes of Full Metal Alchemist is not representative of anime as a whole, nor should it be held as such. there are plenty of anime which hit reset at each new episode as well. For another, it's not as though no Western cartoon follows a set continuity; that stereotype has already been debunked several times in this thread alone.
Ultimately, it boils down to personal tastes. It's not a requirement that comedy cartoons follow a singular continuity; if one does, then that's fine, but it shouldn't be treated as a standard requirement, nor is a comedy show "inferior" for not doing so. As stated previously, comedy and action/drama are 2 different animals.
Personally, I don't mind shows with stand-alone plots and no major episode to episode continuity beyond the fundamentals because they're just easier to follow; with those shows you can watch the episodes in any given order without the flow being interrupted.